Geometry Similar Triangle Theorems Test Solutions and Scoring (PDF)

\(\frac{x}{12} = \frac{10}{15}\)

\(15x = 120\)

\(x = 8\)

\(AB = 8 + 12 = 20\)

x = 8; AB = 20

Scoring: 1 point for the proportion, 1 point for x and AB.

\(\frac{6}{8} = \frac{3}{4}\) and \(\frac{9}{12} = \frac{3}{4}\).

Equal ratios: by the converse of the Triangle Proportionality Theorem, DE is parallel to BC.

Scoring: 1 point for the ratios, 1 point for the conclusion.

1. \(\sqrt{9^{2} + 12^{2}} = 15\)
2. \(9^{2} = 15 \cdot AD\), so \(AD = 5.4\); \(12^{2} = 15 \cdot DB\), so \(DB = 9.6\)
3. \(h = \sqrt{5.4 \cdot 9.6} = \sqrt{51.84} = 7.2\)

Scoring: 1 point per part.

\(\frac{9}{6} = \frac{12}{8} = \frac{15}{10} = 1.5\).

All three pairs are proportional, so the triangles are similar by SSS~ with scale factor 1.5.

Yes, SSS~, k = 1.5

Scoring: 1 point for the ratios, 1 point for the test and the factor.

\(\frac{2x}{x + 6} = \frac{9}{6}\)

\(12x = 9x + 54\)

\(x = 18\)

\(AD = 36\)

x = 18; AD = 36

Scoring: 1 point for the equation, 1 point for x and AD.

Angle A is shared. \(\angle ADE\) and \(\angle ABC\) are corresponding angles of parallel lines, so they are congruent.

Two pairs of congruent angles: similar by AA.

Scale factor \(\frac{4}{10} = 0.4\), so \(DE = 0.4 \cdot 15 = 6\).

AA; DE = 6

Scoring: 1 point for the angles, 1 point for AA, 1 point for DE.

\(\sqrt{6 \cdot 24} = \sqrt{144} = 12\). The altitude is the geometric mean of the segments, so it is also 12.

12; 12

Scoring: 1 point each.

\(\frac{h}{40} = \frac{5}{10}\), so \(h = 20\).

20 feet

Scoring: 1 point for the proportion, 1 point for the height.

The area ratio is \(\left( \frac{4}{10} \right)^{2} = 0.16\).

\(0.16 \cdot 125 = 20\)

20

Scoring: 1 point for the ratio, 1 point for the area.